I will describe recent evidence that anti-de Sitter spacetime is nonlinearly unstable. Generic small but finite perturbations eventually form small black holes. There are special solutions, called geons, which remain nonsingular and do not collapse. I will also describe (weaker) evidence that black holes in anti-de Sitter spacetime are also unstable. The endstate may involve a violation of cosmic censorship. It would be of great interest to check all of these claims using numerical relativity.